Magnetic Field due to Moving Charges

A moving charge generates a magnetic field in addition to its electric field. The magnetic field magnitude is directly proportional to the charge and inversely proportional to the square of the distance from the charge. Unlike the electric field, the magnetic field direction is perpendicular to the plane containing the velocity vector and the line joining the charge to the field point.

The magnetic field magnitude depends on the charge's speed and the sine of the angle between the velocity vector and the line connecting the source to the field point. The field is maximum in the plane perpendicular to the charge velocity, where the angle is 90 degrees. This angular dependence distinguishes magnetic fields from electric fields, which have no directional velocity component.

Use the right-hand rule: point your thumb in the direction of the charge's velocity, and your curled fingers indicate the magnetic field direction. For a negative charge, reverse the field direction. This convention helps visualize the perpendicular relationship between velocity and the resulting magnetic field at any point in space.

The magnetic field is expressed as the cross product of the velocity vector and a unit vector pointing toward the field point, multiplied by the permeability of free space (μ₀ = 4π × 10⁻⁷ T·m·A⁻¹). This equation shows that the magnetic field is proportional to both charge magnitude and velocity, and inversely proportional to the square of the distance.

The Tesla (T) is the SI unit for magnetic field strength, named after inventor Nikola Tesla. One Tesla represents the magnetic field strength produced by a moving charge or current. The unit is derived from the fundamental equation relating magnetic field to charge, velocity, distance, and permeability of free space.

Both fields are proportional to charge magnitude and inversely proportional to distance squared. However, the electric field points radially from the charge, while the magnetic field is perpendicular to both the velocity and the radial direction. Additionally, the magnetic field depends on charge velocity and the angle between velocity and the field point location.

The magnetic field arises from the cross product of velocity and the unit vector toward the field point. By definition, a cross product produces a vector perpendicular to both input vectors. This geometric relationship means the magnetic field can never point along the velocity direction, distinguishing it fundamentally from the electric field produced by the same charge.